def dstep

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/natfuel.bend:401

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/logic.bend as L
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA
import bend-collections-laws-math@1.0.0.0/spec/math/natural.bend as S
import bend-collections-laws-math@1.0.0.0/proofs/math/natural/fact.bend as FA
import bend-collections-laws-math@1.0.0.0/proofs/math/natural/roots.bend as RT

401def dstep(+n: Nat, +s: Nat, +h: {Nat.is_lt(s, n) == True{} : Bool}) -> {Nat.is_le(S.desc(n, s), S.desc(n, 1n+s)) == True{} : Bool}:
402  +pos = L.subst(Nat, z => {Nat.is_lt(0n, z) == True{} : Bool}, 1n+Nat.sub(n, 1n+s), Nat.sub(n, s), Equal.sym(Nat, Nat.sub(n, s), 1n+Nat.sub(n, 1n+s), FA.sub_lt_succ(n, s, h)), {==})
403  L.subst(Nat, z => {Nat.is_le(S.desc(n, s), z) == True{} : Bool}, Nat.mul(S.desc(n, s), Nat.sub(n, s)), Nat.mul(Nat.sub(n, s), S.desc(n, s)), NA.mul_comm(S.desc(n, s), Nat.sub(n, s)), RT.le_mul_pos(S.desc(n, s), Nat.sub(n, s), pos))